Summary: We present a game-theoretical meaning for all vertices in the simple simplex, which are included among the stationary points, of the replicator dynamics of asymmetric two-person games. It is well known that there exists a relationship between a stationary point and a Nash equilibrium. Indeed, the so-called folk theorem of evolutionary game theory claims that a stable stationary point is closely related to a Nash equilibrium in the case of symmetric two-person games. However, for unstable stationary points, its game-theoretical meaning remains unclear. Hence, in this paper, we introduce indices for unstable stationary points by using the Jacobian matrix of the replicator dynamics. We discuss a game-theoretical meaning of the indices, and present an alternative solution concept to the Nash equilibrium of a bimatrix game. Then, any bimatrix game always has this solution if we restrict strategies to pure ones.